Exhaustion by compact sets explained

X is a nested sequence of compact subsets K _{i} of X (i.e. K _{1\subseteq} K _{2\subseteq} K _{3\subseteq …} ), such that K _{i} is contained in the interior of K _{i+1} , i.e. K _{i\subseteqint(K} _{i+1} ) for each i and X = c up ^{inf ty} _{i =1} K _{i} . A space admitting an exhaustion by compact sets is called exhaustible by compact sets.

For example, consider

X = R ^{n} and the sequence of closed balls K _{i} = \{ x : |x| \le i \}.

Occasionally some authors drop the requirement that

K _{i} is in the interior of K _{i+1} , but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets.

Properties

The following are equivalent for a topological space

X :^[1] X is exhaustible by compact sets.X is σ-compact and weakly locally compact.X is Lindelöf and weakly locally compact.(where weakly locally compact means locally compact in the weak sense that each point has a compact neighborhood).

The hemicompact property is intermediate between exhaustible by compact sets and σ-compact. Every space exhaustible by compact sets is hemicompact^[2] and every hemicompact space is σ-compact, but the reverse implications do not hold. For example, the Arens-Fort space and the Appert space are hemicompact, but not exhaustible by compact sets (because not weakly locally compact),^[3] and the set

\Q of rational numbers with the usual topology is σ-compact, but not hemicompact.^[4]

Every regular space exhaustible by compact sets is paracompact.^[5]

References

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. .
• Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. .
• Book: Lee . John M. . John M. Lee . Introduction to topological manifolds . 2011 . Springer . New York . 978-1-4419-7939-1 . 2nd.

External links

• • Web site: Existence of exhaustion by compact sets . Mathematics Stack Exchange.

Notes and References

1. Web site: A question about local compactness and $\sigma$-compactness . Mathematics Stack Exchange.
2. Web site: Does locally compact and $\sigma$-compact non-Hausdorff space imply hemicompact? . Mathematics Stack Exchange.
3. Web site: Can a hemicompact space fail to be weakly locally compact? . Mathematics Stack Exchange.
4. Web site: A $\sigma$-compact but not hemicompact space? . Mathematics Stack Exchange.
5. Web site: locally compact and sigma-compact spaces are paracompact in nLab . ncatlab.org.